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# Construction of the Hamiltonian and Density Operators

We assume the system can be modeled with the Hamiltonian,

 (2.1)

Separately we assume that there exists a bath which can be modeled with the Hamiltonian,

 (2.2)

Finally, as stated before, the interaction of the bath and system is taken in the form of

 (2.3)

The total Hamiltonian is,

 (2.4)

We assume uncertainty in the preparation of states, so we switch to the density operator regime. We call the density operator corresponding to our bath coupled system (S: Single original oscillator; B: Environment) where the individual density operators can be retrieved via a trace, i.e.

 Trace over environment states Trace over local states

The dynamics of the system evolve as (Liouville equation),

 (2.5)

We commit a unitary transform to simplify this equation as follows (the so-called interaction picture). It can be shown that,

 (2.6)

Furthermore, from the premises of Quantum Mechanics, it is legitimate to commit transforms of the type,

 Unitary (2.7)

These so called similarity transformations preserve rank, determinant, trace, and eigenvalues. We let,

 (2.8)

We take the time derivative and find,

 (2.9)

From equation 2.6, since

and ,

 (2.10)

where is the transform of and can be calculated as follows. We know that,

and whereas , equation 2.6 gives

can be calculated in this fashion for both the and the multiple set , where it is thus now trivial to find that,

 (2.11)

If we now define

we can simplify to

 (2.12)

Since the bath is in equilibrium, we can construct easily. Let . Then the probability that one mode (i) of the field is excited with n photons is

 (2.13)

The density matrix for this mode is then
 (2.14)

To simplify this further, we imagine a three states system in which we can represent as, for example letting ,

The above sum could then be written,

We recover the matrix, and so we can write just as well,

 (2.15)

By the rules of probability, the density operator for the entire bath becomes,

 (2.16)

By our assumptions, this is the state of the bath for all time, and so we will not write this as a function of time. The form will take will be much more complicated, and it is the purpose of this report to derive the differential equation defining its time evolution.

Next: The Integration Up: Detailed Derivation of the Previous: Assumptions
Timothy Jones 2006-10-11